Deterministic equivalents for certain functionals of large random matrices

• 成果类型: Article
• 署名作者: Hachem, Walid; Loubaton, Philippe; Najim, Jamal
• 署名单位: Universite Paris Saclay; Centre National de la Recherche Scientifique (CNRS); CNRS - Institute for Information Sciences & Technologies (INS2I); Universite Gustave-Eiffel; ESIEE Paris; Institut Polytechnique de Paris; Ecole Nationale des Ponts et Chaussees; Centre National de la Recherche Scientifique (CNRS); IMT - Institut Mines-Telecom; Institut Polytechnique de Paris; Telecom Paris
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/105051606000000925
• 发表日期: 2007
• 页码: 875-930
• 关键词: limiting spectral distribution; dimensional random matrices; sample covariance matrices; Empirical distribution; eigenvalues

摘要

Consider an N x n random matrix Y-n = (Y-ij(n)) where the entries are given by Y-ij(n) = sigma(ij)(n)/root n X-ij(n) the X-ij(n) being independent and identically distributed centered with unit variance and satisfying some mild moment assumption. Consider now a deterministic N x n matrix A(n) whose columns and rows are uniformly bounded in the Euclidean norm. Let Sigma(n) = Y-n + A(n). We prove in this article that there exists a deterministic N x N matrix-valued function T-n(z) analytic in C - R+ such that, almost surely, [GRAPHICS] Otherwise stated, there exists a deterministic equivalent to the empirical Stieltjes transform of the distribution of the eigenvalues of Sigma(n)Sigma(T)(n). For each n, the entries of matrix T-n(z) are defined as the unique solutions of a certain system of nonlinear functional equations. It is also proved that 1/N Trace T-n(z) is the Stieltjes transform of a probability measure pi(n)(d lambda), and that for every bounded continuous function f, the following convergence holds almost surely. C-n(sigma(2)) = (1)/(N) E log det (IN + Sigma(n)Sigma(T)(n)/sigma(2)), where sigma(2) is a known parameter.